Surds - Finding square roots.

To find square root of surd like this : $a+\sqrt{b}+\sqrt{c}+\sqrt{d}$ We put it equal to $\sqrt{x}+\sqrt{y}+\sqrt{z}$

To find the square root of : $21-4\sqrt{5}+8\sqrt{3}-4\sqrt{15}$ can we put this equal to $\sqrt{x}+\sqrt{y}+\sqrt{z}$ please guide...

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Do you mind if I edit your title to lower case letters? – 1015 Feb 21 '13 at 17:39
It looks like you did not get the memo that typing in all caps is bad internet ettiquette, so I changed that for you. Please remember this for future posts :) Doh, someone beat me to it... – rschwieb Feb 21 '13 at 17:40

Note that $$(a+b\sqrt 3+c\sqrt 5+d\sqrt {15})^2\\=(a^2+3b^2+5c^2+15d^2)+(2ab+10cd)\sqrt3+(2ac+10bd)\sqrt 5 +(2ad+2bc)\sqrt{15}.$$
Do you know if there is a classification of numbers which satisfy that their square root has $a=0$? Saying that the coefficients must be of the form $3b^2+5c^2 + 15d^2, 10cd, 10bd, 2bc$ isn't really helpful since we still have to solve for those values. – Calvin Lin Feb 21 '13 at 18:40